Beyond six feet: The collective behavior of social distancing

In a severe epidemic such as the COVID-19 pandemic, social distancing can be a vital tool to stop the spread of the disease and save lives. However, social distancing may induce profound negative social or economic impacts as well. How to optimize social distancing is a serious social, political, as well as public health issue yet to be resolved. This work investigates social distancing with a focus on how every individual reacts to an epidemic, what role he/she plays in social distancing, and how every individual’s decision contributes to the action of the population and vice versa. Social distancing is thus modeled as a population game, where every individual makes decision on how to participate in a set of social activities, some with higher frequencies while others lower or completely avoided, to minimize his/her social contacts with least possible social or economic costs. An optimal distancing strategy is then obtained when the game reaches an equilibrium. The game is simulated with various realistic restraints including (i) when the population is distributed over a social network, and the decision of each individual is made through the interactions with his/her social neighbors; (ii) when the individuals in different social groups such as children vs. adults or the vaccinated vs. unprotected have different distancing preferences; (iii) when leadership plays a role in decision making, with a certain number of leaders making decisions while the rest of the population just follow. The simulation results show how the distancing game is played out in each of these scenarios, reveal the conflicting yet cooperative nature of social distancing, and shed lights on a self-organizing, bottom-up perspective of distancing practices.

The table contains the equilibrium strategies for a set of distancing games computed based on the general theory on population games, which assumes that the population is large and well mixed, i.e., every individual can interact with all others in the population.Assume that there are n activities, and the activities are independent.Let x = {xi : i = 1, .., n} be the distancing strategy of an individual and y = {yi : i = 1, …, n} the average strategy of the population.Let pi(y) = wisi(yi) be the potential distancing risk at activity i, where wi is the risk factor of activity I and sI is a logistic function.Then, the equilibrium strategy of the game can be obtained with a formula, xi* = sI -1 (l/wi) for all i where l is a constant such that 1 = Si si -1 (l/wi) (details in Methods -Equilibrium strategies and stabilities).
The games are defined for 20 CASA activities.Therefore, n = 20, and wi = di ai + (1 -di) bi, where ai and bi are contact factor and impact factor for activity i, respectively, and di is a severity parameter, 0 £ di £ 1.The same logistic function si is used for all i with ki = 10 and qI = 0.5.The CASA activities and their estimated contact and impact factors are described and given in Table 1 in the main text.By setting di to 0, 0.25, 0.5, 0.75, 1 for all i, five games are defined, and their equilibrium strategies are computed using the above formula.For each game (corresponding to a specific value for dI), the participating frequencies at equilibrium are listed in SR1-Table-1.The frequencies are represented by the active times in hours per week (112 active hours in total).

Results in SR1-Table-2:
This table shows the convergence of the distancing strategies to their equilibrium strategies in small world social networks.The network parameters are fixed for m = 2000, K = 6, and b = 0.3.The neighborhood size is varied from 1 to 2, 3, 4, 5, 6.Five games are simulated with di set to 0, 0.25, 0.5, 0.75, 1 for all i, correspondingly.The strategies obtained from the simulation are compared with their equilibrium strategies.The average Euclidean norms of the differences between the individual strategies and their equilibrium strategies are listed.
The simulation is started with every individual assigned an initial strategy x randomly generated around the corresponding equilibrium strategy x*, i.e., for every individual, and for every activity i, xi* is first perturbed randomly by 20%; then an initial frequency xi is generated randomly within 100% of deviation from the perturbed value of xi*.The simulation then proceeds and terminates, in most cases, in less than 20 generations, when either the average difference between the individual strategy and the equilibrium strategy is small enough or no strategy is improved in a few consecutive generations.
Each simulation is repeated for five times with five different sets of randomly generated initial strategies for all the individuals.The results in the table are the average outcomes.They show that the distancing strategies from the simulation consistently converge to their equilibrium strategies for all the di values and for all neighborhood sizes greater than 1.The converge is not as accurate when the neighborhood size is too small (k = 1).

Results in SR1-Table-3:
This table shows the convergence of distancing strategies to their equilibrium strategies in small world social networks.A single game is simulated with di fixed to 0.5 for all i, the network parameters m fixed to 2000 and K to 6, but the randomness parameter b is changed from 0.1 to 0.2, 0.3, 0.4, 0.5.When b is smaller, the neighborhood of each network node tends to be more local or in other words, the connections among the individuals are shorter-ranged and less random.The game is also simulated with different neighborhood sizes, k = 1, 2, 3, 4, 5, 6.The strategies obtained from the simulation are compared with their equilibrium strategies.The average Euclidean norms of the differences between the individual strategies and their equilibrium strategies are listed.
The simulation is again started with every individual assigned an initial strategy x randomly generated around its equilibrium strategy x*, i.e., for every individual, and for every activity i, xi* is first perturbed randomly by 20%; xi is then generated randomly within 100% of deviation from the perturbed value of xi*.Each simulation is repeated for five times.The results in the table are the average outcomes.
The simulation results do not show significant differences with different levels of randomness of the social network.

Results in SR1-Table-4:
This table shows the convergence of the distancing strategies to their equilibrium strategies in small world social networks.A single game is simulated with di fixed to 0.5 for all i and the network parameters m fixed to 2000, K to 6, and b to 0.3.However, the way to generate the initial strategies is varied: For every individual, and for every activity i, xi* is perturbed randomly by r = 0%, 10%, 20%, 30%, 40%, or 50%; and the initial frequency xi is generated randomly within 100% of deviation from the perturbed value of xi*.
The game is then simulated with the neighborhood size varied from 1 to 2, 3, 4, 5, 6.The strategies obtained from the simulation are compared with their equilibrium strategies.The average Euclidean norms of the differences between the individual strategies and their equilibrium strategies are listed.Each simulation is repeated for five times.The results in the table are the average outcomes.The results show that the simulation would converge as expected if the distribution of the initial strategies is not too far from the equilibrium strategy (r < 50%).

Results in SR1-Figure-1:
This figure contains the snapshots from the simulation for the distancing game in a small world social network.The parameters for the game are fixed to di = 0.5 for all i and r = 20%, and for the network to m = 2000, K = 6, and b = 0.3.There are two columns of plots in the figure.The first one is from the simulation with the neighborhood size k = 1.The second one is from the simulation with the neighborhood size k = 2.
In each column, the first four plots show the changes of the individual strategies in four different generations.In each of the plots, along the x-axis are the 20 CASA activities.Over each activity, there are 2000 circles corresponding to the participating frequencies in this activity by the 2000 individuals.Along the y-axis are the participating frequencies for the activities represented by the active times in hours per week (112 active hours in total).
The last plot in each column shows the changes of the average difference between the individual strategy and the equilibrium strategy.As shown in the first column, for k = 1, the individual strategies are not improved after first few generations.The average difference between every individual strategy and the equilibrium strategy cannot be reduced further after 21 generations when the simulation is terminated.
However, when k = 2, as shown in the second column, the individual strategies converge quickly in several generations.The first plot shows the initial strategies at the 1 st generation when they appear to be quite random; the second plot shows the strategies in the 4 rd generation when they start converging; the third plot shows the strategies in the 7 th generation when they almost converge to their equilibrium positions; the fourth plot shows the strategies in the 10 th generation when they are close enough to their equilibrium values, and the simulation is terminated.As shown in the last plot, the average difference between every individual strategy and the equilibrium strategy is decreased as the generation increases and is eventually reduced to < 10 -4 .Results in SR1-Figure-2 and SR1-Figure-3 are produced in the same way as those in SR1-Figure-1 except that SR1-Figure-2 shows the results for k = 3, 4 and SR1-Figure-3 for k = 5, 6.The results for k = 3, 4, 5, 6 are all similar with those for k = 2, showing that the simulated distancing strategies converge to the equilibrium strategies for all neighborhood sizes greater than or equal to 2.

Table legends :
Act -activities: 1-20 as given in Table1in Section 2.1; di -severity parameter in [0, 1]; cell contents -frequencies: active times in hours per week (112 active hours in total)